About a year after his paper on discrete-data filtering, R. E. Kalman coauthored a second paper with R. S. Bucy on continuous filtering (1). This paper also proved to be a milestone in the area of optimal filtering. Our approach here will be somewhat different from theirs, in that we will derive the continuous filter equations as a limiting case of the discrete equations as the step size becomes small.* Philosophically, it is of interest to note that we begin with the discrete equations and then go to the continuous equations. So often in numerical procedures, we begin with the continuous dynamical equations; these are then discretized and the discrete equations become approximations of the continuous dynamics. Not so with the Kalman filter! The discrete equations are exact and stand in their own right, provided, of course, that the difference equation model of the process is exact and not an approximation.
The continuous Kalman filter is probably not as important in applications as the discrete filter, especially in real-time systems. However, the continuous filter is important for both conceptual and theoretical reasons, so this appendix will be devoted to the basics of continuous filtering.
First, we assume the process and measurement models to be of the form: